In this post, we cover two of the most commonly used option greeks and see how they are actually used. Greeks are always covered extensively when option pricing is taught, but it’s not until I had to send out greeks along with option prices did I get ask, “how did you scale your delta?”
Sounds silly but sometimes greeks can look a bit strange. When your option is deep out-of-the-money (OTM), it can be confusing to tell whether the Delta, for example, is in cents or percent. In this industry, two kind of people can get upset when you send them the “wrong” greeks: someone who’s very picky, or someone with very little knowledge of what you are actually sending them (i.e. someone very junior at your client’s firm or accountants). So to remove some of the confusion, let’s go a bit more into the details on Delta and Vega…
Usually in a Black-Scholes implementation, the delta is computed in just one extra line of code (will have sample code available soon). Specifically:
Delta= Exp( -DivYield * TMat) * N( Sign* d1) * Sign
Sign: Call=1; Put=-1
d1= [Log(S_0/ K) + (Rf - DivYield + 0.5 * Sigma^2) * TMat]/ [Sigma * Sqrt(TMat)]
N(.) is the normal Gaussian distribution’s cumulative probability function
The delta here is a measure of the probability of moneyness or the rate of change in option price per unit change in the underlying’s price.
Okay, that’s all good. But on a daily Mark-to-Market (MTM) or Profit/ Loss (PnL) basis, what’s a trader most interested in? That would be the so called “Trader’s Delta”, dollar change in option price per percent change in the underlying (i.e. how much would be option move if my underlying moved 1%?). This Trader’s Delta is slightly different from the delta that’s a measure of probability of moneyness. Namely:
Trader’s Delta= Delta * S_0 * 1%
For example, consider a Brent Crude Oil Jun 2012 90 Put with Spot a 110.27, volatility of 33%, and 114 calendar-days until maturity. The options is deep out-of-the-money (OTM) have a delta of 11.59%, which means if Brent Crude goes down a dollar, my option goes up $0.1159. The option price is approximately $1.24, so that would be a good 9.35% return on your position, if the underlying only move 0.91%. The “Trader’s Delta” or Dollar Delta would be 11.59% * 110.27 * 1%= $0.1277.
What about Vega?
Vega is also a one liner addition in your standard Black-Scholes implementation:
Vega= 0.01 * Exp( -DivYield * TMat) * N’( d1) * S_0 * Sqr( TMat)
N’(.) is the normal Gaussian distribution’s probability density function
The Vega given here is very straight forward: dollar change in option price per 1 volatility point. Using our Brent Jun 2012 90 Put example, the Vega would be $0.1203. Meaning if volatility goes up one vol. point to 34%, our option goes up $0.1203, a 9.67% return on your option position.
In this put example, it seems that our Delta, at $0.1277, is bigger than our Vega, $0.1203, both being on dollar terms. However, it is not often that Brent Jun 2012 futures will drop 1% a day. Volatility, on the other hand, can easily move up or down 1 or 2 vol. points a day. Therefore, buying a deep OTM option is almost as much a bet on implied volatility as the underlying’s directional bet.
From looking at greeks and PnL on large portfolios of options across all asset class daily, a very frequently observed behaviour is that implied volatility and the underlying spot price is inversely correlated. That means put options will tend to generate more profit in a down turn than a call option could in an up turn.
January 20, 2012 No Comments
The bread and butter of pricing anything, is yield curve building. The risk-free curve is what you use for drifting Equities forward in the “risk-neutral” measure, it is used for modelling expected floating coupons in the future, and for discounting. Technical discussion of what the risk-neutral measure is would be address later in another post, or if the reader can kindly Google the term (There are many very technical and very detail discussion of the risk-neutral measure written by esteemed academics, I will not go there but could provide a summary later in layman’s terms).
Suppose we would like to price a simple interest rate swap (IRS), fixed vs floating, we would need a yield curve to do so. To build a yield curve, we need data. Cash rates, Interest rate futures, forward rate agreements (FRA), and swap rates are some of the most common data point to be used for bootstrapping yield curves (I will also post later on the quirks and quarks of bootstrapping, which may sound simple but could get very complicated very quickly when the type of rate we want to model get more and more complicated).
I do recall that school and some textbooks teach us to use T-Bill, T-Note rates, or Zero-Coupon Bonds (ZCB). This is only, if you are specifically building a curve to price T-Bill, T-Notes, or ZCB. But in practice, we don’t use those kind of instruments for pricing for two major reasons:
- Liquidity: T-bill, T-Note, ZCB, some could argue are very liquid. However, it is definitely not as liquid as Swap rates, when we are considering bootstrapping the long end of the curve for major G10 currencies. Swap rates are traded literally around the clock by every broker/ bank.
- Libor: the second reason Swap rates is used as the default curve building instrument is because from it we bootstrap Libor, the most commonly referenced rate for IR derivatives (IRD). T-bill, T-note, and ZCB rates are not Libor. Priced into them are also the credit risk of the issuer.
The standard swap traded have the following conventions:
During curve building, the forward rate that we bootstrapped has the same tenor as the Floating Leg frequency. So with a standard 3M USD-Libor curve that you build, for example, you may think that the forward 6M rate can be stripped. But in practice it will be wrong, because of Basis Risk. That’s why Basis Swaps, for example, 3M vs 6M Libor, exists because the spread between 3M and 6M libor are stochastic and cannot be captured deterministically (i.e. from just a 3M curve or 6M curve).
Basis swap spreads are useful for deriving curves for rates of non-standard tenor. The same can be achieved by getting swap rates where the floating frequency matches the tenor of the rate you are trying to model. The tenor of the rate one tries to model also affect what instrument should be used for the middle part of the curve:
- For 3M rates: use IR Futures, which are mainly traded on 3M forward rate (convexity adjustment required, but it’s usually really small impact)
- For 6M rates: use FRAs, which are mainly traded on various forward start tenor on 6M forward rates
So suppose we have a 3M floating, non-standard, EUR swap, to model the floating coupons we would use a 3M Euribor accruing curve. However, what is used for discounting might be bit of a shocker! In the industry, we use the standard tenor curve (i.e. 6M Euribor) as the discounting curve. So for a 6M floating EUR swap, the accruing and discounting curve is the same. The choice of the discounting curve I think goes back to what risk-neutral measure one is in (i.e. what’s the risk-free asset). Recently a bit more confusion are created by a shift toward using domestic Overnight-Index Swap (OIS) curve for discounting Cross-Currency swaps. This will be covered in a follow-up post. To conclusion, a distinction has been made between the accruing and the discounting curve and what instruments/ data to use when building either curve specifically for the purpose of pricing IRD.
January 16, 2012 No Comments
I read an interesting post on FT Alphaville yesterday on the Dividend Sandwich http://ftalphaville.ft.com/blog/2011/04/12/543891/how-to-make-a-dividen d-sandwich/> . From my understanding the strategy goes like this:
1. Long Div Futures
2. Short Div Futures Options (Call)
Looks like a covered call to me. The investor is long dividend (purchased at the price of the div futures) and limited his upside at the strike of the div futures option.
The author, Theo Casey, made one point in this post that were wrong in my opinion.
“Unlike volatility, this is not based solely on what is implied. It’s also based on what is realised, i.e. forecasts and sentiment only matter to a point. The value of the contracts is ultimately anchored by what is paid.”
From my understanding, Volatility Swap, and the more commonly traded, Variance Swap, trade on realized Volatility/ Variance. In a seasoned Vol/Var Swap, like Div Futures/ Swap, what is realised is actually very important.
His post points out several good observations:
1. There are no pricing model for Dividend Futures Options. As far as I know, most people use a simple Black Scholes model to price these stuff. Dividends are sticky, however, so there are lot of model risk here. Given the size of the market at 2B euro (the futures market is at 8B euro), I suspect the market makers are just rolling in the profit and it will be a while before the banks will release their model.
2. It was also mentioned that the market is approaching fair value and writing puts now maybe profitable (hinting at a mis-price on the downside implied dividend vol).
Maybe interesting to see what a Div Futures vol surface look like…. maybe good idea for another post
My conclusion: Dividend Sandwich sounds exotic but is nothing new. Whichever way you spin it, it is nothing more than a covered-call or married-put on Dividend. But much love to Theo for bringing the attention to this new asset class that’s attracting lots of interest recently.
April 14, 2011 No Comments
Nothing too exciting but nonetheless:
Types of arbitrage strategies Many of the physical commodity markets in which Glencore operates are fragmented or periodically volatile. As a result, discrepancies generally arise in respect of the prices at which the commodities can be physically bought or sold in different geographic locations or time periods, taking into account the numerous relevant pricing factors, including freight and product quality. These pricing discrepancies can present Glencore with arbitrage opportunities whereby Glencore is able to generate profit by sourcing, transporting, blending, storing or otherwise processing the relevant commodities. Whilst the strategies used by Glencore’s business segments to generate such margin vary from commodity to commodity, the main arbitrage strategies can be generally described as being:
* Geographic – where Glencore leverages its relationships and production, processing and logistical capabilities in order to source physical commodities from one location and deliver them to another location, where such commodities can command a higher price (net of transport and/or other transaction costs);
* Product-related – where it is possible to exploit the blending or multi-use characteristics of the particular commodities being marketed, such as the various crude oil products, coal or concentrates, in order to supply products which attract higher prices than their base constituents, or exploit existing and/or expected price differentials; and
* Time-related – where it is possible to exploit a difference between the price of a commodity to be delivered at a future date and the price of a commodity to be delivered immediately, where the available storage, financing and other related costs until the future date are less than the forward pricing difference.
Glencore uses market information made available by its marketing and industrial teams across its many locations to identify arbitrage opportunities. Glencore’s marketing and investment activities and relationships with producers and consumers of raw materials are supported by a global network of more than 50 offices providing sourcing and distribution capabilities located in over 40 countries throughout Europe, North, Central and South America, the CIS, Asia, Australia, Africa and the Middle East. This network provides Glencore with visibility over shifting supply and demand dynamics in respect of significant volumes of physical commodities across the globe.
The detailed information from Glencore’s widespread operations and close relationships with producers, consumers and logistics providers is available to Glencore’s marketing operations and often enables them to identify opportunities, taking into account Glencore’s extensive logistics capabilities, to source and supply physical commodities at attractive margins.
Link to the Glencore document is here: http://www.glencore.com/documents/Glencore_ITF_14_April_2011.pdf
April 14, 2011 No Comments
In case you don’t know, there’s a new movie out call Margin Call http://www.margincallmovie.com/which is “a methodical, coolly absorbing boardroom thriller set on the eve of the 2008 economic collapse. Unfolding over a tense 36-hour period at a Wall Street investment firm where drastic damage-control measures are afoot”.
It premiere at Sundance and should be out sometime later this year. Here’s the Trailer:
April 12, 2011 No Comments
Here’s a nice paper by Optiver that explains High Frequency Trading
April 7, 2011 No Comments
Couple weeks again I had a post on the Virtual Economy and…
Now it’s official, Korean Supreme Court have ruled that Virtual Curency has real currency value. Here’s the link to the news report. The ruling follows from the appeal of two gamers who were acquited for “illegally” exchanging game currency for Korean Won. So the “illegal” activity should now technically be called “arbitrage” (although the supreme court upheld the fines from the original court’s ruling).
This ruling should have huge implication and quite possibly many unintended consequences. The obvious implication is that “Virtual Currency Market” could emerge very soon and obviously there would be lots of arbitrage opportunities. However, the bigger economic implication is rather more difficult to grasp. What is the “real” good that’s being produced in the Virtual Economy?
January 20, 2010 No Comments
Found an interest read from Bloomberg today about “Floored”, a documentary about pit traders. While most consider it a dying art, I digged up this old link from PBS’s own documentry, “Open Outcry”, which has a section on hand signal and here’s another link. Computers, quants, and programmers have their advantages but I think that most in the investment professions have been aspired by the excitement of the trading pits in one form or another.
Here’s the link to the official website of “Floored”
January 19, 2010 No Comments
Can you name the country with a minmum wage of $3.42/hour, GNP of $2,266 per capita, and 77th richest in the world?
… EverQuest anyone?
January 12, 2010 No Comments
MIT’s Technology Review ran a good article of High Frequency trading. Here’s an interesting observation from Dr. Wilmott:
The increasing dominance of algorithmic trading and the growing speed of execution, he says, could cause tiny price changes to snowball, rolling down the hill at exponentially increasing speed–either because the machines are trading too fast or because too many funds are trading in the same style. “The potential is there for a crash to happen quite quickly,” he says.
Which begs the question, what’s the unintended consequences of high frequency trading? Many argues that HFT is there to take advantage of the little guys. But afterall HFT is nothing more than a system, a highly rule based system, and like any system invented, there’s doom to be a loophole some where. Can High Frequency Trading be gamed?
Also found this video from Reuters: Revenge of the Nerds
December 23, 2009 6 Comments